A078971 Numbers n such that C(4n,n)/(3n+1) (A002293) is not divisible by 4.

0, 1, 3, 5, 11, 13, 21, 43, 45, 53, 85, 171, 173, 181, 213, 341, 683, 685, 693, 725, 853, 1365, 2731, 2733, 2741, 2773, 2901, 3413, 5461, 10923, 10925, 10933, 10965, 11093, 11605, 13653, 21845, 43691, 43693, 43701, 43733, 43861, 44373, 46421, 54613

Offset: 1

Author

Benoit Cloitre, Jan 14 2003

Keywords

• nonn

Comments

Stanica observes that the sequence in binary forms a pattern where 1 bits are inserted into the word 1010101...:

1 11

101 1011 1101

10101 101011 101101 110101

1010101 10101011 10101101 10110101 11010101...

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..5050
• P. Stanica, p^q Catalan numbers and squarefree binomial coefficients, arXiv:math/0010148 [math.NT], 2000.

Crossrefs

Cf. A000225 (C(2n, n)/(n+1) is not divisible by 2), A003462 (C(3n, n)/(2n+1) is not divisible by 3), A003463 (C(5n, n)/(4n+1) is not divisible by 5).

Programs

Magma

[n: n in [0..2*10^4] | not IsZero(Binomial(4*n,n) div (3*n+1) mod 4)]; // Vincenzo Librandi, Apr 16 2015

Mathematica

Select[ Range[0, 65000], Mod[ Binomial[4#, # ]/(3# + 1), 4] != 0 &] (* Robert G. Wilson v, Oct 12 2005 *)

PARI

isok(n) = binomial(4*n,n)/(3*n+1) % 4; \\ Michel Marcus, Apr 16 2015

Python

from _future_ import division
A078971_list = []
for t in range(100):
    A078971_list.append((2**(2*t)-1)//3)
    for j in range(t):
        A078971_list.append((2**(2*t+1)+2**(2*j+1)-1)//3) # Chai Wah Wu, Mar 06 2016

Extensions

Comments and more terms from Ralf Stephan, Oct 30 2003

a(28)-a(44) from Robert G. Wilson v, Oct 12 2005